Flows of singular vector fields and applications to fluid and kinetic equations

Several physical phenomena arising in fluid dynamics and kinetic equations can be modeled by nonlinear transport PDE. Such quantities are the vorticity of a fluid, or the density of a collection of particles advected by a velocity field which is highly irregular. The theory of characteristics provides a link between this PDE and the ODE dX/dt=b(t,X(t,x)), where b is the velocity field. When b has Sobolev or BV regularity and bounded divergence, the theory of DiPerna-Lions and Ambrosio gives a good notion of solution to the ordinary differential equation using the concept of regular Lagrangian flow. Extending the results of Crippa-DeLellis, and more recently Bouchut-Crippa, we study Lagrangian flows associated to velocity fields with anisotropic regularity: those with gradient given by the singular integral of an L^1 function in some directions, and the singular integral of a measure in others. We exploit an anisotropic version of the previous arguments and estimate the difference quotients in this context, thereby gaining quantitative estimates in terms of the given regularity bounds. One then recovers well-posedness for the ordinary differential equation. This answers positively the question of existence of Lagrangian solutions to the Vlasov Poisson and Euler equations with L^1 data.